Hypercomputation

Hypercomputation is the theory of methods for the computation of non-recursive functions. The classes of functions which they can compute is studied in the field known as recursion theory.

Hypercomputation was first introduced by Alan Turing in his 1939 paper Systems of logic based on ordinals, which investigated mathematical systems in which an oracle was available to compute a single arbitrary (non-recursive) function from naturalss to naturals.

Other posited kinds of hypercomputer include:

• A quantum mechanical system which somehow uses (for example) an infinite superposition of states to compute a non-recursive function¹. Such a system could not be an ordinary qubit quantum computer.
• A Turing machine which is running for an infinite period of time (perhaps the observer is being dropped into a black hole).
• A Turing machine which is accelerating on exponentially short time scales (in a Newtonian universe, such a gadget might operate by manufacturing a clone of itself which was only half the size and operated at twice the speed).
• A non-deterministic Turing machine which has a preference ordering over its final states.
• An analog computer might be able to perform hypercomputation if physics admits real variables (not just computable reals), and these are in some way "harnessable" for computation. This might require quite outlandish laws of physics (for example, a measurable physical constant with the value &Omega).

At this stage, none of these devices seem physically plausible, and so hypercomputers are likely to remain a mathematical fiction.

See also: the Church-Turing thesis.

Notes

1. There have been some claims to this effect; see Tien Kieu, Quantum Algorithm for Hilbert's Tenth Problem and the ensuing literature. It is very likely that these results will turn out to be erroneous or non-physical. Until this is well established and explained, the possibility of quantum hypercomputation is deserving of investigation.

References

1. Alan Turing, Systems of logic based on ordinals, Proc. London math. soc. 45, 1939
2. Tien Kieu, Quantum Algorithm for the Hilbert's Tenth Problem, Physics e-print archive quant-ph/0110136v2
3. Toby Ord, Hypercomputation: computing more than the Turing machine, Honours Thesis, University of Melbourne, 2002.